You should've used the word morha9a (مُرهَقة), ta3baana (تَعبانة) is only used in certain dialects and not in MSA.
No, مرهقة would better be translated "exhausted." متعبة (mut3aba[-tun]) is the word to use here.
Does the statements "my sister is tired" and "my tired sister" differ in Arabic?
Yes, they do:
my sister is tired = أُختي مُرهَقة (Literally "My sister tired")
my tired sister = أُختي المُرهَقة (Literally "My sister the tired")